Existence of entropy solution for a nonlinear parabolic problem in weighted Sobolev space via optimization method

    Lhoucine Hmidouch   Affiliation
    ; Ahmed Jamea   Affiliation
    ; Mohamed Laghdir   Affiliation


This paper investigates the existence result of entropy solution for some nonlinear degenerate parabolic problem in weighted Sobolov space with Dirichlet type boundary conditions and L1 data.

Keyword : nonlinear parabolic problem, opimization method, Dirichlet type boundary, entropy solution, weighted Sobolev space

How to Cite
Hmidouch, L., Jamea, A., & Laghdir, M. (2023). Existence of entropy solution for a nonlinear parabolic problem in weighted Sobolev space via optimization method. Mathematical Modelling and Analysis, 28(3), 393–414.
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Sep 4, 2023
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A. Abbassi, C. Allalou and A. Kassidi. Existence of entropy solutions for anisotropic elliptic nonlinear problem in weighted Sobolev space. In The International Congress of the Moroccan Society of Applied Mathematics, pp. 102–122. Springer, 2019.

S. Antontsev and S. Shmarev. Anisotropic parabolic equations with variable nonlinearity. Publicacions Matematiques, pp. 355–399, 2009.

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. Vázquez. An l1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 22(2):241–273, 1995.

V. Bhuvaneswari, S. Lingeshwaran and K. Balachandran. Weak solutions for p-Laplacian equation. Advances in Nonlinear Analysis, 1(4):319–334, 2012.

L. Boccardo and Th. Gallouët. Non-linear elliptic and parabolic equations involving measure data. Journal of Functional Analysis, 87(1):149–169, 1989.

A. Cavalheiro. Weighted Sobolev spaces and degenerate elliptic equations. Boletim da Sociedade Paranaense de Matema´tica, 26(1-2):117–132, 2008.

A. Cianchi and V. Maz’ya. Second-order regularity for parabolic p-Laplace problems. The Journal of Geometric Analysis, 30(2):1565–1583, 2020.

P. Drabek, A. Kufner and V. Mustonen. Pseudo-monotonicity and degenerated or singular elliptic operators. Bulletin of the Australian Mathematical Society, 58(2):213–221, 1998.

G. Gagneux and M. Madaune-Tort. Analyse mathématique de modéles non linéaires de l’ingénierie pétroliére, volume 22. Springer Science & Business Media, 1995.

A. El Hachimi, J. Igbida and A. Jamea. Existence result for nonlinear parabolic problems with l1-data. Appl. Math.(Warsaw), 37(4):483–508, 2010.

J. Heinonen, T. Kilpelanen and O. Martio. Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover Publications, Inc., Mineola, New York, 2006.

L. Hmidouch, A. Jamea and M. Laghdir. Optimization method for a nonlinear degenerate parabolic problem in weighted Sobolev spaces. Discussiones Mathematicae: Differential Inclusions, Control & Optimization, 41(1), 2021.

A. Khaleghi and A. Razani. Existence and multiplicity of solutions for p(x)Laplacian problem with Steklov boundary condition. Boundary Value Problems, 2022(1):1–11, 2022.

M. El Ouaarabi, A. Abbassi and C. Allalou. Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaces. Journal of Elliptic and Parabolic Equations, 7(1):221–242, 2021.

M. El Ouaarabi, A. Abbassi and C. Allalou. Existence and uniqueness of weak solution in weighted Sobolev spaces for a class of nonlinear degenerate elliptic problems with measure data. International Journal of Nonlinear Analysis and Applications, 13(1):2635–2653, 2022.

M. El Ouaarabi, C. Allalou and A. Abbassi. On the Dirichlet problem for some nonlinear degenerated elliptic equations with weight. In 2021 7th International Conference on Optimization and Applications (ICOA), pp. 1–6. IEEE, 2021.

M. El Ouaarabi, C. Allalou and S. Melliani. Existence of weak solution for a class of p(x)-Laplacian problems depending on three real parameters with Dirichlet condition. Bolet´ın de la Sociedad Matem´atica Mexicana, 28(2):1–16, 2022.

M. Ragusa, A. Razani and F. Safari. Existence of radial solutions for p(x)Laplacian Dirichlet problem. Advances in Difference Equations, 2021(1):1–14, 2021.

K. Rajagopal and M. Ružička. Mathematical modeling of electrorheological materials. Continuum mechanics and thermodynamics, 13(1):59–78, 2001.

M. Růžička. Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Math, 1748:16–38, 2000.

T. Singer. Existence of weak solutions of parabolic systems with p, q-growth. manuscripta mathematica, 151(1):87–112, 2016.

M. Tsutsumi. Existence and nonexistence of global solutions for nonlinear parabolic equations. Publications of the Research Institute for Mathematical Sciences, 8(2):211–229, 1972.

B.O. Turesson. Nonlinear potential theory and weighted Sobolev spaces, volume 1736. Springer Science & Business Media, 2000.

N. Weisheng, M. Qing and C. Xiaojuan. Asymptotic behavior for nonlinear degenerate parabolic equations with irregular data. Applicable Analysis, 100(16):3391–3405, 2021.

M. Xu and S. Zhou. Existence and uniqueness of weak solutions for a generalized thin film equation. Nonlinear Analysis: Theory, Methods & Applications, 60(4):755–774, 2005.

C. Zhang. Entropy solutions for nonlinear elliptic equations with variable exponents. Electronic Journal of Differential Equations, 92(2014):1–14, 2014.

C. Zhang and S. Zhou. A fourth-order degenerate parabolic equation with variable exponent. J. Partial Differ. Equ, 22(4):376–392, 2009.

V. Zhikov. On density of smooth functions in Sobolev–Orlich spaces. Zapiski Nauchnykh Seminarov POMI, 310:67–81, 2004.